While \(S_2\)  contains important structural information, prior work has established that microstructures with a specific \(S_1\)  and \(S_2\)  are highly degenerate [1]}, [2]}, [3]}, [4]}, [5]}. Furthermore, the set of \(S_1\)  and \(S_2\)  degenerate microstructures is infinitely large in the thermodynamic limit. This degeneracy implies that the other microstructural descriptors of these two-phase systems will generally differ. There is a variety of descriptors that incorporate higher-order information that one could consider to differentiate \(S_2\)  degenerate microstructures [6]}. At first glance, a natural higher-order function to include beyond \(S_1\)  and \(S_2\)  is the three-point function \(S_3\) . However, Jiao, Stillinger, and Torquato revealed that \(S_3\)  does not appreciably increase information content over pair statistics in systems that lack long-range order [2]}.
